An eigenvector of a square matrix is a direction that the matrix does not turn. Applying the matrix to it produces the same vector scaled by some factor, and that factor is the eigenvalue. This calculator finds both for 2x2 and 3x3 matrices, and prints the characteristic polynomial it solved so the answer can be checked rather than merely accepted.
Arb Digital built this page with one specific behaviour in mind: complex eigenvalues are real results and are displayed as complex numbers, not as an error or a blank. A rotation matrix has no real eigenvalue because no direction survives a rotation unturned, and a calculator that returns "NaN" for that case is hiding correct mathematics behind a bug-shaped message. When the discriminant goes negative here, you get the conjugate pair written out properly.
What This Eigenvalue Calculator Does
It builds the characteristic polynomial det(A − λI), solves it exactly using the quadratic formula for 2x2 matrices and a closed-form cubic solution for 3x3, and then finds an eigenvector for each real eigenvalue by computing the null space of A − λI. Alongside the roots it reports the trace, the determinant, the spectral radius — the largest absolute eigenvalue — and whether the roots are real and distinct, repeated, or a complex conjugate pair.
The determinant machinery underneath is the same as our determinant calculator, which is worth reading first if the expansion itself is unfamiliar: this page differs by solving det(A − λI) = 0 for λ rather than evaluating a determinant at fixed entries. For plain matrix arithmetic, use the matrix calculator instead.
How to Use It
- Choose 2x2 or 3x3. The characteristic equation is a quadratic in the first case and a cubic in the second.
- Type the matrix row by row. Order matters — eigenvalues are a property of the matrix, not of the set of numbers in it.
- Read the eigenvalues. Real roots are listed in descending order; a complex conjugate pair is written in the form a ± bi.
- Check the characteristic polynomial. It is printed below the grid so you can verify the coefficients against your own working.
- Read the eigenvectors. Each is given for its eigenvalue and scaled to convenient whole-ish numbers, since any non-zero multiple is equally valid.
The Formula and How It's Calculated
An eigenvalue λ satisfies Av = λv for some non-zero vector v. Rearranged, that is (A − λI)v = 0, which has a non-zero solution only when A − λI is singular — that is, when det(A − λI) = 0. Expanding that determinant produces the characteristic polynomial, whose roots are the eigenvalues.
For a 2x2 matrix the polynomial is always λ² − (trace)λ + (determinant) = 0, and the quadratic formula finishes the job. For a 3x3 it is λ³ − c₂λ² + c₁λ − c₀ = 0, where c₂ is the trace, c₁ is the sum of the three principal 2x2 minors, and c₀ is the determinant of the whole matrix.
Work through the default. The matrix has rows (2, 0, 0), (0, 3, 4) and (0, 4, 9). Its trace is 14. The three principal minors are 6, 18 and 11, summing to 35. Its determinant is 22. So the characteristic equation is λ³ − 14λ² + 35λ − 22 = 0, whose roots are 11, 2 and 1. Check them against the identities: 11 + 2 + 1 = 14, matching the trace, and 11 × 2 × 1 = 22, matching the determinant. The eigenvector for λ = 2 is (1, 0, 0), for λ = 11 it is (0, 1, 2), and for λ = 1 it is (0, 2, −1). Wolfram MathWorld's eigenvalue entry gives the general definition this follows.
Complex Eigenvalues Are Not an Error
Enter the 2x2 matrix with rows (0, −1) and (1, 0) — a ninety-degree rotation. Its characteristic equation is λ² + 1 = 0, and the eigenvalues are +i and −i. There is no real answer because rotating the plane leaves no direction pointing the way it started, so no real vector can be merely scaled by the transformation. The maths is not failing; the geometry genuinely has no real eigenvector.
Complex eigenvalues always arrive in conjugate pairs for a real matrix, and they carry a clean interpretation. Write the pair as a ± bi: the magnitude √(a² + b²) is the scaling factor per application, and the angle atan2(b, a) is the rotation per application. A pair with magnitude below 1 spirals inward, a pair with magnitude above 1 spirals outward, and a pair sitting exactly on the unit circle rotates forever without growing. This is why oscillating systems and rotations produce complex spectra as a matter of course, and why a tool that returns NaN here is discarding the most informative part of the answer. Our complex number calculator handles arithmetic on the pair once you have it.
What Eigenvalues Tell You About a System
Repeatedly applying a matrix is the standard model for anything that evolves in discrete steps — a population moving between age classes, a web surfer moving between pages, a state machine stepping forward. In that setting the eigenvalues are the growth rates along each independent direction, and the largest one dominates everything after enough steps.
That gives a compact stability rule. If every eigenvalue has absolute value below 1, repeated application shrinks any starting vector towards zero and the system is stable. If any eigenvalue exceeds 1 in absolute value, its direction grows without bound and eventually swamps the rest. Exactly 1 is the boundary case, and it is why the dominant eigenvalue of a Markov transition matrix is always exactly 1 — total probability is conserved rather than growing. The spectral radius reported in the grid is precisely this dominant magnitude, and it is usually the single most useful number on the page.
Eigenvectors Have No Fixed Length
If v is an eigenvector then so is 2v, and so is −v, and so is 0.001v. Eigenvectors specify a direction, not a particular arrow, so any non-zero scalar multiple is exactly as correct. This calculator scales each one to a tidy form with a leading positive entry where possible, but a textbook, a spreadsheet and a numerical library will each present a different multiple of the same direction — most numerical libraries normalise to unit length, which produces decimals like 0.4472 and 0.8944 instead of 1 and 2.
The practical consequence is that "my answer does not match" is usually not an error. Divide one vector by the other component by component: if every ratio is the same number, the two answers are the same eigenvector. If the ratios differ, then something really is wrong. The sign is the most common difference of all, since −v is always as valid as v.
Repeated Eigenvalues and Defective Matrices
When a root of the characteristic polynomial appears twice, one of two things is true. Either the repeated eigenvalue has two independent eigenvectors, in which case the matrix behaves normally and can still be diagonalised — the identity matrix is the extreme case, where every vector is an eigenvector. Or it has only one, in which case the matrix is called defective and cannot be diagonalised at all.
The classic defective example is the 2x2 matrix with rows (1, 1) and (0, 1). Its only eigenvalue is 1, repeated, and its only eigenvector direction is (1, 0). There is no second independent direction to find, and any tool that reports two distinct eigenvectors there is inventing one. This calculator flags a repeated root explicitly rather than silently printing the same vector twice, because the distinction changes what you can legitimately do next. Defective matrices need a Jordan form rather than a diagonalisation, which is beyond the scope of this page.
Symmetric Matrices Are the Well-Behaved Case
If a matrix equals its own transpose, a great deal is guaranteed in advance. All its eigenvalues are real — no complex pairs are possible. Eigenvectors belonging to different eigenvalues are automatically perpendicular to one another. And the matrix can always be diagonalised, even when eigenvalues repeat, so the defective case cannot arise.
This is why symmetric matrices turn up wherever the underlying quantity is a covariance, a distance or an energy: those objects are symmetric by construction, so the analysis is guaranteed to behave. Principal component analysis is exactly the eigendecomposition of a covariance matrix, with each eigenvalue giving the variance captured along its component and each eigenvector giving the direction of that component. The default matrix on this page is symmetric, which is why its three eigenvalues come out real and its eigenvectors come out mutually perpendicular. MIT OpenCourseWare's 18.06SC Linear Algebra course devotes a full unit to why symmetry forces these properties, and Gilbert Strang's differential equations series shows the same eigenvalues driving continuous systems.
Arb Digital builds data tooling and analytics where every calculation is shown, checked and reproducible rather than hidden in a black box.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Treating a complex pair as a failure — a negative discriminant is a correct result describing rotation, not an error to be worked around.
- Expecting eigenvectors to match another tool exactly — any non-zero multiple of an eigenvector is the same eigenvector, and libraries normalise differently.
- Forgetting the sign convention in the characteristic polynomial — for a 3x3 the λ² coefficient is the trace with a minus sign attached, and dropping it changes every root.
- Assuming a repeated eigenvalue always has two eigenvectors — defective matrices have fewer independent directions than eigenvalues and cannot be diagonalised.
- Skipping the trace and determinant check — the eigenvalues must sum to one and multiply to the other, and that test catches most arithmetic slips instantly.
Related Free Tools From Arb Digital
Evaluate a determinant on its own with the determinant calculator, invert a matrix with the matrix inverse calculator, solve simultaneous equations with the system of equations calculator, find polynomial roots with the cubic equation calculator, or work with individual vectors using the vector calculator. The free online tools hub lists every maths calculator we publish.
Frequently Asked Questions
It is the factor by which a matrix stretches one particular direction. If applying the matrix to a vector produces the same vector scaled by three, then three is an eigenvalue and that vector is its eigenvector.
It is the determinant of the matrix minus lambda times the identity, expanded as a polynomial in lambda. Setting it to zero and solving gives the eigenvalues.
Because the transformation includes a rotation, so no real direction is left merely scaled. Complex eigenvalues are a correct answer, and for a real matrix they always come in conjugate pairs.
Yes. A zero eigenvalue means the matrix collapses at least one direction to nothing, which also means the determinant is zero and the matrix has no inverse.
Any non-zero multiple of an eigenvector is also an eigenvector. Most numerical libraries scale theirs to unit length, so the components look different while the direction is identical.
Yes, counting repeats and including complex ones. They also multiply to the determinant, and both identities are useful checks on any hand calculation.
The same scaling factor applies to more than one direction, or the matrix is defective and has fewer independent eigenvectors than eigenvalues. The second case cannot be diagonalised.
This page explains a mathematical calculation for educational purposes only, and results should be checked independently before being relied on in engineering or academic work.
